Percentage Change Over Multiple Periods: Chaining, CAGR and Averages
A single percentage change is easy. The trouble starts the moment you have two or more of them in a row: prices that rose three years running, a salary adjusted every January, traffic that fell one quarter and recovered the next. Almost everyone's first instinct — add the percentages together — is wrong, and the error grows the bigger the numbers get. This guide shows the correct method, the arithmetic behind it, and how to express a multi-period change as one clean annual figure.
Multiply growth factors, never add percentages. Convert each change into a factor (+8% becomes 1.08, −5% becomes 0.95), multiply them together, then subtract 1. Three years of +10%, −5% and +8% give 1.10 × 0.95 × 1.08 = 1.1286, i.e. +12.86% overall — not the +13% you get by adding.
Turn every change into a factor
A growth factor is simply 1 plus the change expressed as a decimal. It converts "change" language into "multiply by this" language, which is what compounding actually does.
| Change | Factor | Applied to 200 |
|---|---|---|
| +25% | 1.25 | 250 |
| +8% | 1.08 | 216 |
| 0% | 1.00 | 200 |
| −5% | 0.95 | 190 |
| −40% | 0.60 | 120 |
Once every period is a factor, combining periods is pure multiplication — no special cases for increases versus decreases.
Chaining several periods
The overall factor is the product of the individual factors:
Total factor = F₁ × F₂ × … × Fₙ, and total change = (total factor − 1) × 100.
Suppose a subscription price moved +6%, +6% and +6% over three renewals. Adding gives 18%. Multiplying gives 1.06³ = 1.191, so the real increase is 19.1%. On a $40 plan that is $47.64, not $47.20 — small here, but the gap widens fast with larger rates or more periods.
Why +10% then −10% is not zero
Because the second percentage is taken from a different base. Start at 100: a 10% rise puts you at 110, and a 10% fall from 110 removes 11, not 10, leaving 99. As factors: 1.10 × 0.90 = 0.99, a 1% net loss. Reverse the order and you get exactly the same 0.99 — order never changes the product, but it does change the intermediate values.
The size of the shortfall grows with the square of the rate. Up 50% then down 50% leaves 0.75 — a 25% loss. Up 90% then down 90% leaves 0.19. This asymmetry is why a portfolio that drops 50% needs a 100% gain to break even.
You never come back to where you started, and the shortfall grows with the square of the rate.
The average annual rate (CAGR)
To express a multi-year change as a single steady rate, take the n-th root of the total factor:
CAGR = (End ÷ Start)1/n − 1, where n is the number of periods.
Revenue of $400,000 growing to $650,000 over four years: 650,000 ÷ 400,000 = 1.625; 1.6250.25 = 1.1291; CAGR ≈ 12.9% per year. Note that this is the geometric average, not the arithmetic one. Averaging the yearly percentages would overstate growth whenever the rates vary — the more volatile the series, the bigger the overstatement.
A worked five-year example
A shop's annual sales, with each year's change:
| Year | Change | Factor | Sales |
|---|---|---|---|
| Start | — | — | $500,000 |
| 1 | +12% | 1.12 | $560,000 |
| 2 | −8% | 0.92 | $515,200 |
| 3 | +20% | 1.20 | $618,240 |
| 4 | +3% | 1.03 | $636,787 |
| 5 | −2% | 0.98 | $624,051 |
Total factor: 1.12 × 0.92 × 1.20 × 1.03 × 0.98 = 1.2481, so sales rose 24.81% over five years. The arithmetic average of the yearly changes is (12 − 8 + 20 + 3 − 2) ÷ 5 = 5.0%, but the true CAGR is 1.24810.2 − 1 = 4.54%. The half-point gap is the volatility drag, and it is why fund factsheets quote annualised rather than average returns.
Four common mistakes
- Adding percentages across periods. Only valid as a rough estimate when every rate is tiny (under about 2%) and the number of periods is small.
- Averaging percentages instead of annualising. Always overstates growth for a fluctuating series.
- Mixing bases. Percentage change always uses the value at the start of that period as the denominator, not the original starting value.
- Confusing percent with percentage points. A rate moving from 4% to 6% is a rise of 2 percentage points — and a 50% increase. See percent vs percentage points.
Practice problems
1. Rent rises 4%, then 6%, then 5%. What is the total increase?
2. A stock falls 30%, then rises 30%. Where does it stand?
3. Users grow from 12,000 to 30,000 in three years. What is the CAGR?
4. What single change is equivalent to −20% followed by −25%?
Frequently Asked Questions
Related reading: how to calculate percentage increase (the single-period case behind every chain), the percentage formula, percentages in finance and percentage difference vs change.
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